- . N = Σ m = . Table : . HM N m Σ Special applications of Harmonic Mean The Harmonic Mean is restricted in its field of usefulness.
It is useful for computing the average rate of increase of profits of a concern or average speed at which a journey has been performed or the average price at which an article has been sold. The rate usually indicates the relation between two different types of measuring units that can be expressed reciprocally. For example, if a man walked 20km in hours, the rate of his walking speed can be expressed - - Descriptive statistics and probability Example . An automobile driver travels from plain to hill station 100km distance at an average speed of 30km per hour.
He then makes the return trip at average speed of 20km per hour what is his average speed over the entire distance (200km)? If the problem is given to a layman he is most likely to compute the arithmetic mean km hours km per hour where the units of the first term is a km and the unit of the second term is an hour or reciprocally, hours km hour per km where the unit of the first term is an hour and the unit of the second term is a km. of two speeds. i.e., X km km kmph But this is not the correct average.
So harmonic mean would be mean suitable in this situation. Harmonic Mean of and is HM kmph = + = . . Relationship among the averages In any distribution when the original items differ in size, the values of AM, GM and HM would also differ and will be in the following order AM GM HM ≥ ≥ If all the numbers X , X , …, X n are identical then, AM = GM = HM.
Example . Verify the relationship among AM, GM and HM for the following data Xf log X f log X f / X . . .
N / = = fX / = log / = . / = . Table : . - - AM fX N